A mass spring system is in equilibrium. If I pull on the load by $x$ meters, the energy stored in the spring is (this is what is given in my book):
$$E=\frac12kx^2 $$
However, doesn't the load lose gravitational potential energy as it moves down? Where would this energy go? By conservation law, shouldn't the energy equation be:
$$E_{stored}= \frac12kx^2 + mgx$$
In short, where does the loss of gravitational potential energy (mgx) of the load get transferred to if it is not stored in the spring?
(Referring to a vertical mass spring system)
The picture in my mind:
Best Answer
There are some very interesting subtleties here. Let's analyze the situation very carefully.
Let's choose our system to consist of the block, spring, and Earth. By choosing the Earth and block to be in our system, we will have a change in gravitational potential energy.
In the beginning, the (massless) spring hangs vertically with a block of mass $m$ attached at the bottom. We could calculate how much the spring is stretched by equating the gravitational and spring forces ($kx_1=mg$) but we won't need this.
Now, during the pulling process you describe, it's important to note that you are doing positive work on the system, which means that the energy in the system increases. It is tempting to say that the change in energy is zero, but this isn't the case for the system we've chosen.
Let's use the work-energy theorem to answer your question of where the gravitational potential energy "goes." $$\underbrace{W_\text{net, external}}_\text{Positive}=\Delta E_\text{tot}=\underbrace{\Delta U_\text{grav}}_\text{Negative}+\underbrace{\Delta U_\text{elastic}}_\text{Positive}$$
Yes, the gravitational potential energy decreases. Where does it go? Well, the only other term that could (mathematically) compensate for this decrease in gravitational potential energy is the increase in elastic potential energy. But be careful with wording here. The spring is not storing gravitational potential energy; rather, gravitational potential energy was converted to elastic potential energy.
As a side note, since the left-hand side of the equation above is positive, the absolute value of $\Delta U_\text{elastic}$ is greater than that of $\Delta U_\text{grav}$. So, not only was the gravitational potential energy converted to elastic potential energy, the positive work you did on the system also adds to the increase in elastic potential energy.